To see whether the first order behavior we observe is an expected result, we performed a Pisarski-Wilczek [1]} stability analysis of the
chiral transition in the multirep theory.
The calculation is an analysis of the critical behavior of an effective three-dimensional field theory of the two chiral condensates of the theory.
If any infrared-stable fixed points exist for this theory, the transition can be second order if it occurs in the basin of attraction of that fixed point (but may still be first order otherwise).
If no infrared-stable fixed points exist, the transition must be first order.
To proceed, we first identify the symmetries and spontaneous symmetry-breaking (SSB) pattern of our theory
and use them to construct the most general Landau-Ginzburg-Wilson (LGW) Lagrangian, including
only relevant and marginal terms.
We compute the \(\beta \)  functions for this effective theory to one loop using the
\(\epsilon \)  expansion: we expand as usual in small \(\epsilon = 4-d\) , but then set \(\epsilon =1\) .
Finally, we perform a stability analysis of the resulting \(\beta \)  functions in the zero-mass
theory (with relative signs of couplings constrained to induce the correct SSB pattern).
The validity of the calculation is limited by both the \(\epsilon \)  expansion and by working to only one loop.
Analyses which go to higher order in perturbation theory and treat three-dimensionality more cautiously can find stable fixed points that are missed by the approach used here [2]}, [3]}.
